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Transport and vortex pinning in micron-size
superconducting Nb films
Lamya Ghenim, Jean-Yves Fortin, Gehui Wen, Xixiang Zhang, Claire
Baraduc, Jean-Claude Villegier
To cite this version:
Lamya Ghenim, Jean-Yves Fortin, Gehui Wen, Xixiang Zhang, Claire Baraduc, et al.. Transport and
vortex pinning in micron-size superconducting Nb films. Physical Review B: Condensed Matter and
Materials Physics (1998-2015), American Physical Society, 2004, 69, pp.064513. �hal-00002859�
ccsd-00002859, version 1 - 16 Sep 2004
Lamya Ghenim∗
Institut Laue-Langevin, B.P. 156, 38042 Grenoble, France and CNRS
Jean-Yves Fortin†
CNRS, Laboratoire de Physique Th´eorique, UMR7085, 3 rue de L’Universit´e, 67084 Strasbourg Cedex, France
Wen Gehui, Xixiang Zhang
Department of Physics, HK University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
Claire Baraduc, Jean-Claude Villegier
D´epartement de Recherche Fondamentale sur la Mati`ere Condens´ee, CEA-Grenoble, F-38054 Grenoble cedex 9, France.
(Dated: September 16, 2004)
We have carried out Hall measurements on thin films of Nb in the flux flow regime. The Hall bars were several microns in scale. Oscillations with magnetic field in the tranverse and longitudinal resistances between the depinning field Bdand the upper critical field Bc2 are observed below Tc.
The Hall effect may even change sign. The tranverse and longitudinal resistances are interpreted in terms of current-driven motion of vortices in in the presence of a few impurities. Simulations from time-dependent Ginzburg-Landau equations (TDGL) confirm this argument.
PACS numbers:74.25.Qt,74.78.Db,47.32.Cc,74.78.-w
The conductivity in the flux-flow regime of type-II superconductors is determined by the dynamics of vor-tices. Since vortex motion in an applied external mag-netic field is intrinsically at least two dimensional, to understand the transport the full conductivity tensor is needed [1, 2, 3]. In past studies of vortex matter [4], the current-voltage curves have been observed to have in-teresting effects such as aging and memory phenomena. Together with effects of hysteresis in the magnetization curves, these have been interpreted as the consequences of the motion of interacting vortices in a complex en-ergy landscape with random potential. This provides a general explanation of memory effects, slow relaxation rates and sensitivity to initial conditions. In this pa-per we present expa-perimental evidence of similar memory and pinning effects in Hall and resistance measurements of type-II thin films, using samples of micron dimension well below Tc. By relating these results to simulations
of the dynamics of vortices interacting with one another and a vortex scattering potential, we will deduce a length
∗present address: DSV/DRDC Laboratoire Biopuces- Bˆat
4020, CEA-Grenoble, 17, rue des Martyrs, 38054 Grenoble, email:ghenim@dsvsud.cea.fr
†email:fortin@lpt1.u-strasbg.fr
scale for the vortex pinning potential.
Up to now, observed anomalies in the Hall effect close to Tc in bulk samples have been explained as being
re-lated to vortex motion damping [5] or plastic flow of the vortex lattice [6]. As we diminish the sample size, one would expect to reach a regime where the motion of a finite number of vortices must be considered. In order to simplify full vortex dynamics, in which one must consider motions of the vortex lines, we have fabricated planar samples with thickness Lz = 800 − 1000˚A, comparable
to the coherence length ξ at 4K. In this case the vortex core can be considered as a disk. We took a conventional superconductor, Nb, with samples as pure as possible to minimize pinning. Nb has the advantage that the bulk vortex structure has been studied in detail by neutron scattering [7] and microscopy [8]. The epitaxial samples (100) were grown by DC magnetron sputtering on R-plane sapphire at 600o
C and the eight-lead Hall bars pro-cessed by photolithography and reactive-ion etching (the bar area is Lx× Ly= 50µm×5µm; the resistance probes,
which are micron wide, are separated by Lp = 10µm).
DC Hall resistance Rxyand magnetoresistance Rxx
mea-surements were made in the flux-flow regime with current reversal in the B transverse configuration. To eliminate the effects of contact misalignment, the Hall resistance
2
was obtained by subtracting the positive and negative magnetic field data. The films had Tc = 8.4K and a
re-sistive ratio ρxx(300K)/ρxx(10K) = 5.1 with a carrier
density ne≃ 8.5 × 1022cm−3, obtained from the
expres-sion Rxy = B/eneLz for the Hall resistance in the
nor-mal phase. The low resistance contacts on the Hall bars were made by Indium bonding. The measurements were made with a Quantum Design Magnetic Property Mea-surement System adapted for transport meaMea-surements, giving high stability as a function of magnetic field and temperature. Figure 1 shows magnetoresistance as a function of magnetic field at T = 4K, well below Tc at
current I = 200µA. The resistance Rxxis zero for fields
below Bd, the depinning field for vortices, and then
in-creases to reach its normal state value at Bc2. From
the upper critical field experimental value Bc2(4K) ≃
1.3T, we use the Ginzburg-Landau theoretical expression Bc2(T ) = φ0/(2πξ
2
(T )), where φ0is the flux quantum, to
deduce approximatively the zero temperature coherence length ξ0 ≃ 115 ˚A. From the carrier density ne, the zero
temperature London penetration length can be estimated using the expression λ0 = pme/µ0nee2, which gives
λ0≃ 180 ˚A, where meis the electron mass, and we obtain
κ ≃ 1.6. The electronic mean free path le is computed
from the normal resistance Rxx at 4K using the Drude
formula le≃ (1.264 × 104Ω)n−2/3e Lp/(RxxLzLy)[9]. The
value of the normal resistance at 4K is Rxx ≃ 1.7Ω,
based on Fig. 1 data, which gives le≃ 76˚A at this
tem-perature, of the same order as the cohence length. The relatively large value of lefor these thin films shows the
good quality of the samples. The transverse resistance displays a more striking behaviour; it oscillates strongly between the depinning field Bd and Bc2. Above Bc2 the
Hall resistance recovers the behavior of the normal state: it is linear with field, with slope inversely proportional to the Nb carrier density. In the field range where Rxy
oscillates, this resistance may even be negative. As seen in Fig. 2, in general a minimum of Rxy corresponds to
a maximum of dRxx/dB . If Rxy were dominated by
a parasitic resistive component, this would tend to give maxima in Rxywhere the oscillating part of Rxxis
maxi-mal, ie where dRxx/dB is between extrema. Subtraction
of positive and negative magnetic field Hall data was im-portant to avoid such parasitic effects. We note that to observe the oscillations the magnetic field had to be swept very slowly: typically at least 12 hours per scan for a given temperature.
6000 10000 14000 18000 H(Oe) 0 0.001 0.002 0.003 0.004 R xy(Ω) Rxx(kΩ)
FIG. 1: Resistances for a sample size Lp ×Ly=10µm×5µm,
T=4K, I=200µA. For convenience Rxxis scaled by a factor
10−3. 5000 15000 H(Oe) 0 0.001 0.002 0.003 0.004 0.005 dRxx/dH (scale 20/3) Rxy
FIG. 2: Comparison of Rxyand dRxx/dH. The parameters
are the same as for Figure 1.
Figure 3 shows Rxy versus B for 2 different
tempera-tures. As we heat the sample (from 4K) the oscillations in Rxy diminish as T approaches Tc over the narrowing
field range between the depinning and the upper critical fields. Measurements at different longitudinal currents are shown in Fig. 4, from which we conclude that the oscillations are a low current phenomena: at 800µA they have almost disappeared. At low currents, the sign may be inverted (compare the data for 100 and 200µA). In fact we observe inverted curves in different field scans at the same current (see insert of Fig. 4). Between these
0 4000 8000 12000 16000 20000 H(Oe) -0.001 0 0.001 0.002 0.003 0.004 Rxy(Ω) T=6K T=4K
FIG. 3: Transverse resistance for different temperatures, I=200µA. 6000 8000 10000 12000 14000 H(Oe) -0.006 -0.004 -0.002 0 0.002 0.004 Rxy(Ω) 200µΑ 800µΑ 100µΑ 6000 8000 10000 12000 -0.002 0 0.002 0.004
FIG. 4: Transverse resistance for different currents, T=4K. In insert, different scans for the same current 200µA, showing history dependence.
scans the sample is kept at low temperature but was sub-ject to different currents and fields. What is surprising is that while the oscillating part of the Hall conductivity is inverted, the pattern is very similar. The sign inver-sion invites comparison to the anomalous sign change observed in the Hall resistance near Tc in the bulk [5],
described phenomenologically in terms of a vortex veloc-ity which has a component opposite to the direction of the superfluid flow. Zhu et al. [6] related the sign rever-sal of the mixed-state Hall resistivity close to the critical temperature to thermal fluctuations and vortex-vortex
interactions and associated this inversion of sign to in-coherent motion of vortices, i.e. plastic flow of vortex lattice. Here the sign reversal is of an oscillating trans-verse Hall effect and is most visible at low temperatures rather than as a smooth change of sign close to Tc, but
we retain from the comparison that it must be related to the dynamics of interacting vortices. P eriodic sharp oscillations in the resistance have been observed in su-perconducting Nb films with a square lattice of artificial pinning centers and are associated with commensurabil-ity of the vortex and antidots densities [10, 11]. Thus the oscillations could be due to pinning. The inversions we see indicate that for a fixed configuration of defects the overall sign depends on the initial vortex configuration. The two striking features of our results: sharp oscilla-tions and sensitivity to initial condioscilla-tions, lead us to pro-pose a dynamical model including both vortex interaction and a quenched pinning potential. In order to describe the evolution in time of the local superconducting and normal flows, and local induced magnetic field, we use the TDGL equations [12, 13, 14, 15, 16, 17]. They are time- and space- dependent nonlinear differential equa-tions coupling the superconducting order parameter and the vector potential and are useful to predict qualitative effects of vortex dynamics in the mixed phase of type-II material. The TDGL equations are derived from the ex-tremum of the superconductor free energy in presence of an external magnetic field Be perpendicular to the
sam-ple: F = a2|ψ|2 +b 4|ψ| 4 + 1 2ms ¯h i∇ − qsA ψ 2 + 1 2µ0 B2 −µ1 0 Be.B (1)
ψ(r, t) is the order parameter of the superconducting phase and we choose a gauge where the scalar potential is zero. The coefficient a is proportional to (T − Tc) and
is negative in the superconducting region. b does not de-pend on the temperature and is positive. ms= 2meand
qs= 2qeare the mass and charge of the Cooper pair. The
steady state solution in the mixed phase in the absence of current is an Abrikosov vortex array. The interactions between vortices are included inside the non linear terms of the differential equations. Inside the vortex cores the magnetic field is maximum and superconducting density minimum. When a homogeneous current J is applied to one direction the vortex array moves in the
trans-4
verse direction in the absence of pinning. The current is introduced as a boundary condition. It is the sum of the contribution from the normal and superfluid compo-nents: J = ˆσE + Js, Js= qs/msℜψ (−i¯h∇ − q¯ sA) ψ.
The conductivity tensor ˆσ is the inverse of the classical resistivity tensor ˆρ. The uniform equilibrium value of ψ [Eq. (1)] in the absence of a magnetic field can be sim-ply written as ψ0 = p−a/b, with a = −a′(1 − T/Tc),
a′ is a positive constant. a′ and b are related to the
zero temperature coherence and penetration lengths by ξ0 = ¯h/
√
2msa′ and λ20 = msb/(µ0q 2
sa′). Within the
do-main of applicability of the TDGL theory, all material parameters can be reduced to the following units:
lengths L → L/ξ0, (2)
temperature T → T/Tc,
magnetic field B → B/Bc2(0),
potential vector A → A/Bc2(0)ξ0
wavefunction ψ → ψ/ψ0
This allows one to study numerically general features of Nb films with only a few dimensionless parameters like κ or reduced temperature t = T /Tc. The time τ is
defined in units of τ0= µ0κ 2
ξ2
0σnnn(T )/ne, where nn is
the normal electron density, and σn the normal regime
conductivity. The dimensionless equation of motion for ψ reads: ∂ψ ∂τ = 1 η " − 1i∇ − A 2 ψ + (1 − t)1 − |ψ|2ψ # , (3)
where η is a relaxation rate proportional to the product of a dimensionless constant me/(κ2¯hµ0σn) and a
numer-ical value. This numernumer-ical value can be estimated from the Bardeen-Cooper-Schrieffer theory[18, 19]. Taking the experimental values (see Table 1), the dimensionless con-stant is close to 20. In references[15, 19, 20], η varies from 0.8 to 12, and we will choose its numerical value around unity to optimize time convergence. From the Maxwell equations we obtain: ∂A ∂τ = ˆρ −κ 2 ∇ × (∇ × A) +(1 − t)ℜ ¯ ψ 1 i∇ − A ψ , (4)
where ρx,x = ρy,y = 1 and ρy,x = −ρx,y = αB are the
normal state components of the resistivity tensor in
di-mensionless units. The coefficient α = σnBc2/qene is
small (α ≃ 10−3 with the values of Table 1). In the
following, we choose κ = 2 (to be clearly type II and close to the experimental value) and T= 0. Equations 3 and 4 are discretized on a grid of size N ξ0× Nξ0 with
time step ∆τ = 0.04. It ensures that the equations converge to a unique steady solution. Taking different time steps around this value does not change the numer-ical results. Instead of using a discretized version of the vector potential Ai,j(τ ), we consider the link variables
Ui,jµ (τ ) = exp(−iAµ,i,j(τ ) aµ), µ = x, y, which preserve
the gauge invariance properties of the continuous model [16, 21]. The resulting equations are accurate to second order in space and time steps. From the time dependence of the link variables, we obtain the instantaneous electric fields and the time averaged resistances.
We calculate with an external field Beperpendicular to
the sample and a bulk current J along the x axis, in units of J0 = qe¯hns(T )/(2meξ0), with ns(T ) = ne− nn(T )
the superconducting electron density. At zero tempera-ture, given the experimental values (Table 1), this gives a current density J0 ≃ 6.85 × 1013A m−2. The
wave-function and magnetic field are periodic along the x axis and ψ vanishes on boundaries of the transverse direc-tion, where the magnetic field takes the values Be± ∆B,
∆B = JLy/2κ2 being the contribution from the current
introduced here as a boundary condition. At the begin-ning of the numerical simulation (t = 0 and Be = 0),
we take as initial conditions ψ = 1 in the bulk of the sample and small random values around zero for the po-tential vector components Ax and Ay. After an
equili-bration time, a few thousand time steps N0, we switch
on the current and measure the resistances, over a pe-riod of time τ = Ntτ0 for each value of the external field
Be, where N0and Ntare the number of iterations of the
equations. Every time we increase the external magnetic field by a small amount, we let the system approach the steady state during N0 time steps before recording the
resistance values.
We model the “defects” in the sample (grain bound-aries, thickness variations, the specific geometry....) by considering a square lattice of points where either the wavefunction vanishes ψ ∼ 0, or it is fixed to a non-zero value depending on the superfluid density ψ ∼ |1 − B/Bc2|. In the first case the impurities decrease the
local condensate density and attract vortices while in the second case they repel. Because the pinning potential is
periodic, it is characterised by a distance between centers denoted by Limp. Limp is the distance between vortex
scattering centers, to be distinguished from le which is
the distance between elastic scattering centers for elec-trons. For simplicity, we choosed a periodic structure for the impurities instead of a random one. This is in order to study the influence of the impurity density, consid-ered as a single parameter, on the voltage oscillations. In the simulations Limp varies from 3.33 (36 impurities,
N = 20) to 12.5 (1 impurity, N = 25). In Fig. 5 we show the simulations for two different impurity concen-trations together with the zero impurity case, shown to indicate the noise in the calculations. The oscillations in the transverse resistance, which are similar for the two forms of pinning potential, strongly resemble those seen in the experiments. From the numerical data, and from the range of system size studied (N=20-25 coherent lengths), we can extract a characteristic period δB, or equivalently a length Lc≡ 1/
√
δB, and plot it as a func-tion of Limp, as shown in the insert of Fig. 5. It is seen
to decrease with increasing Limp, i.e. δB increases with
increasing purity. From the dominant period of the ex-perimental oscillations (Fig. 1) we can extract from the insert of Fig. 5 an effective distance for vortex pinning Limp ≃ 15 − 20ξ0 = 0.17 − 0.23µm. We use ξ0 = 115˚A
and assume we can apply the numerically derived rela-tion between Limp and Lc to our experimental sample.
Indeed, the curve Lc(Limp) depends on the parameters
of the equations 3 and 4, in particular κ, the temperature and the ratio between the system size and the coherent length. Strictly speaking, the TDGL theory may not be accurate in the field range where we extracted Lc, but we
expect to obtain a reasonable estimate of Lcfor κ around
the experimental value 1.6.
In Fig. 6 we calculate Rxy for two different currents:
just as in the experiment (Fig. 4) if the current increases, the oscillation amplitude decreases. Furthermore we ob-serve sign inversion due to differing initial conditions as shown in Fig. 7. The simulations then reproduce the ex-perimental observation of a strong memory effect: while there is correlation in the positions of extrema, the nature of each extremum (maximum or minimum) depends on the initial condition at Bd, but is conserved through the
scan. These simulations, which do not have the geometric features of the contacts, show that the effects seen exper-imentally are properties of generic small devices, and are not just an effect of the specific Hall bar geometry. It is
0.2 0.4 0.6 0.8 1 1.2 Be/Bc2 -0.005 0 0.005 Rxy 4 impurities 16 impurities no impurity 0 5 10 15 20 Limp 2 3 4 5 6 7 8 9 Lc
FIG. 5: Calculated transverse resistances for different impu-rity densities. In insert is Lc versus Limp in units of the
coherence length ξ0. Parameters, whose definitions are in the
text, are N = 25, J = 0.03, Nt= 50000, and η = 0.5.
0.2 0.4 0.6 0.8 1 Be/Bc2 -0.001 0.001 0.003 0.005 Rxy J=0.04 J=0.08
FIG. 6: Calculated transverse resistances for 2 different cur-rents. Parameters are: N = 20, 9 impurities, Nt = 50000,
and η = 0.8.
not excluded, however, that the boundaries of the device contribute to the pinning potential.
In conclusion, we have shown that micron scale Hall bars of thin films display strong oscillations in the trans-port in the flux-flow regime, in particular in the Hall voltage at low current. We have shown numerically that these oscillations may be explained by the effects of pin-ning potentials. The transverse voltage is proportional to the average vortex velocity in the longitudinal direction, which is much smaller than the velocity transverse to the
6 0.5 1 Be/Bc2 -0.005 0 0.005 Rxy data A data B
FIG. 7: Sign inversion seen for different random initial con-ditions A and B. Parameters are: N = 20, 4 impurities, J = 0.03, Nt= 50000, and η = 0.8.
current (Rxy/Rxx≃ 10−3). Rxy is therefore sensitive to
the presence of bulk impurities which act as scattering centers for the vortices. As the transverse component of the vortex velocity is much larger than the longitudi-nal component, a small change in the velocity will have a much larger relative effect on its longitudinal compo-nent, hence Rxy. We explain the sign inversion seen
ex-perimentally (Fig. 4) and numerically (Fig. 7) by the fact that each vortex can be scattered by an impurity in two opposite directions, depending on its initial coordi-nates. The effect is amplified by the collective behavior of other vortices which tend to follow the same direction due to the stiffness of the vortex array. The surprising memory effect, that the inversion continues for the whole field scan, is reproduced by our model of quenched im-purities, at least for the lower field range B/Bc2 <∼ 0.6.
The abnormal Hall effect here is due to pinning of cor-related vortices [22] and not to the dynamics of a single vortex. Thus even potentials of short range, as used in the simulations, are sufficient to influence the transport. Low current transport measurements are then a useful probe of vortex pinning potentials. These potentials are responsible for the complex behavior, as seen here and in other studies of bulk samples [4] that also showed aging and memory phenomena in the current-voltage caracter-istics for example. From the experimental oscillations (Fig. 1) and assuming the caracteristic curve Lc versus
Limpis accurate for the samples studied, we extracted an
effective distance Limp≃ 0.17 − 0.23µm between pinning
centers, for ξ0 = 115˚A, coherent with the large
ampli-tudes seen. Our device is small enough and clean enough that there are a small number of effective pinning cen-ters. More generally we can argue that in micron and, by extension, submicron superconducting devices, strong oscillations in the transport are to be expected as generic properties.
We thank Ping Ao, Guy Deutscher, Philippe Nozi`eres and Tim Ziman for helpful discussions and one of us (L. Ghenim) thanks the Physics Department of the HK Uni-versity of Science and Technology for hospitality.
Table 1
Parameters Experimental Values Lx× Ly× Lz 50µm×5µm×1000˚A Tc 8.4K ne 8.5×1022cm−3 λ0 180˚A ξ0 115˚A κ 1.6 Bc2 1.3T (4K) Rxx (normal state) 1.7Ω (4K) σn (normal state) 1.2 × 107Ω−1m−1 (4K) le 76˚A(4K)
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